390 ON THE JACOBIAN SEXTIC EQUATION. [776 
where ag + 9bf— 20d 2 = 0. It is to be shown, that this implies a single relation between 
the four invariants A, B, C, and A of the sextic function. 
I call to mind that the general sextic has five invariants A, B, G, D, E of the 
orders 2, 4, 6, 10, 15 respectively; the last of them E is not independent, but its 
square is equal to a rational and integral function of A, B, G, D; and instead of D, 
we consider the discriminant A which is an invariant of the same order 10. The 
values of A, B, G are given, Table Nos. 31, 34, and 35 of my Third Memoir on 
Quantics, Phil. Trans., vol. cxlvi. (1856), pp. 627—647, [144]; those of D, A, E were 
obtained by Dr Salmon, see his Higher Algebra, second ed. 1866, where the values of 
A, B, G, D, A, E are all given; only those of A, B, G, A are reproduced in the third 
edition, 1876. 
It may be remarked, that for the general form we have A = ag — 6bf + lace — 10d 2 , 
and that B is the determinant 
a, b, c, d : 
b, c, d, e 
c, d, e, f 
d, e, f, g 
G and A are complicated forms, the latter of them containing 246 terms. But writing 
c = 0, e = 0, there is a great reduction; we have 
A = 
B = 
C = 
2) = 
ag + 1 
ad 2 g — 
1 
a 2 d 2 g 2 + 
1 
a 5 g r> + 
1 
bf - 6 
bf 2 + 
1 
,,df 3 + 
4 
a 4 bfg i 
30 
d 2 - 10 
bdf- 
2 
a bdfg + 
12 
„ dY - 
300 
d i + 
1 
»d*g - 
20 
„ dpg - 
2500 
a°b 3 dg 2 + 
4 
,,/ 6 
3125 
,,bf 3 + 
8 
a 3 bf 2 f — 
15 
„ vdf*- 
24 
„ bdfg 2 - 
4800 
» bdf + 
24 
„ bdf 4 g - 
7500 
,,d 6 - 
8 
„ d*g* + 
30000 
,,df 3 g + 
50000 
a 2 b 3 dg 4 — 
2500 
,,b 3 /y - 
410 
„b*dfy- 
171300 
„ b 2 dp - 
240000 
„ bdfg 2 + 
780000 
„ bdf 4 + 
1200000 
,, dhf - 
1000000 
„ df 2 - 
1600000 
a Pdfg 3 — 
7500 
„ bf 4 g - 
11520 
„ b 3 d 3 g 3 + 
50000 
„ b 3 df 3 g + 
83200 
a°¥g 4 — 
3125 
„ b 5 df 2 g 2 - 
240000 
»bf 5 - 
331776 
„Vdfg 2 + 
1200000 
,,b 4 df 4 + 
1843200 
,,b 3 dy - 
1600000 
„Pdf 3 - 
2560000